Question

Change \(y=\log _{5} x\) to an equivalent statement involving an exponent.

Short answer

\( 5^{y} = x \)

Step-by-step solution

Identify the logarithmic form

The given equation is in logarithmic form: \( y = \log_{5} (x) \). Logarithmic form expresses the power to which the base must be raised to get the number.

Convert to exponential form

To convert a logarithmic equation to an exponential equation, use the property that \( \log_{b} (a) = c \) is equivalent to \( b^{c} = a \). Here, \( b \) is the base (5), \( c \) is the logarithm (y), and \( a \) is the result (x).

Write the equivalent exponential statement

Based on the identified values, rewrite the equation: \( y = \log_{5} (x) \) becomes \( 5^{y} = x \).

Key concepts

logarithmic equations

A logarithmic equation involves the logarithm of a variable quantity. Logarithms are the inverses of exponents. They help us solve equations where the unknown is an exponent. Understanding the form and properties of logarithmic equations is crucial.
For example, in the logarithmic equation \(y = \log_b(x)\), \log_b\ represents the logarithm with base \b\. The equation states that \y\ is the power to which \b\ must be raised to produce \x\.

If you know the base and the logarithm, you can find the number using the properties of logarithms and exponents.

exponential equations

Exponential equations involve variables in the exponent. Converting logarithmic equations to exponential form often simplifies solving them.
For instance, the equation \(y = \log_5 (x)\) can be converted to its equivalent exponential form \[5^y = x\].
This shows the relationship between logarithms and exponents: the exponent is the power to which the base must be raised to get the number.
To convert a logarithmic equation to an exponential equation, remember the general property: \(\log_b(a) = c \) is equivalent to \ b^c = a \.

logarithmic properties

Logarithmic properties are essential for manipulating and solving logarithmic equations. These properties include:

• Product Property: \ \log_b (mn) = \log_b (m) + \log_b (n) \
• Quotient Property: \ \log_b (m/n) = \log_b (m) - \log_b (n) \
• Power Property: \ \log_b (m^n) = n \log_b (m) \

These properties help simplify complex logarithmic expressions and solve logarithmic equations. For example, converting \(\log_5(x) \) to exponential form \[5^y = x\] uses understanding of the relationship between logarithms and exponents.

base of logarithm

The base of a logarithm is the number that is raised to a power to obtain a given number. In the equation \(y = \log_5 (x)\), the base is 5.
This means \y \ is the exponent to which the base 5 must be raised to get \ x \.
Choosing an appropriate base is crucial. Common bases include 10 (common logarithms) and \e\ (natural logarithms used in complex calculations). Knowing the base helps in converting between logarithmic and exponential forms.

equivalent equations

Equivalent equations are different forms of the same equation that express the same relationship. For example, \(y = \log_5 (x)\) and \[5^y = x\] are equivalent.
They show the same relationship between \ y \ and \ x \ using logarithmic and exponential forms respectively.
Converting between them often simplifies solving the equations. If stuck with a logarithmic equation, try converting it to exponential form using properties such as \( \log_b (a) = c \) is equivalent to \ b^c = a \logan equivalent exponential statement helps solve related problems easily.\
Understanding equivalent transformations is vital in mathematics.